Jin Liang, Yunyi Mu, Ti-Jun Xiao
Abstract:
This article concerns evolution equations involving \(\psi\)-Hilfer fractional
derivative in a Banach space. By using the theory of fractional calculus and
\(\psi\)-Laplace transform, we firstly derive a definition of mild solutions for these
equations. Then we establish theorems for the existence and uniqueness of solutions
and the approximate controllability (not the exact controllability) of the
\(\psi\)-Hilfer fractional differential system under appropriate conditions.
We focus on the approximate controllability rather than the exact
controllability because the exact controllability cannot be achieved generally for
the system in infinite-dimensional spaces.
We present a new multidimensional Gronwall-type inequality with multiple singular
kernels involving exponential factors, which extends essentially many existing results.
We also use the new Gronwall-type inequality to study the dependence of the solution
on the order and the initial condition for the fractional integro-differential equations
involving \(\psi\)-Hilfer fractional derivative. Finally, an example is given to
illustrate our main results.
Submitted October 11, 2025. Published November 18, 2025.
Math Subject Classifications: 34K37, 34A08, 45G05, 45E05.
Key Words: psi-Hilfer; mild solution; approximate controllability; Gronwall-type inequality.
DOI: 10.58997/ejde.2025.109
Show me the PDF file (398 KB), TEX file for this article.
| Jin Liang School of Mathematical Sciences Shanghai Jiao Tong University Shanghai 200240, China email: jinliang@sjtu.edu.cn |
| Yunyi Mu School of Arts and Sciences Shanghai Dianji University Shanghai 201306, China email: muyy@sdju.edu.cn |
| Ti-Jun Xiao Shanghai Key Laboratory for Contemporary Applied Mathematics School of Mathematical Sciences Fudan University, Shanghai 200433, China email: tjxiao@fudan.edu.cn |
Return to the EJDE web page